Simplifying Complex Rational Expression

Question

My classmates and I are all struggling to come to an answer on this one complex rational expression.

$$\begin{aligned}\frac{10x^4y}{5x^3y^3}\cdot\frac{25x^2y}{8x^5y^7}\cdot\frac{x^4y^2}{2x^5y^3}&=\\ \frac{2x}{y^2}\cdot\frac{25}{8x^3y^6}\cdot\frac{1}{2xy}&=\\ \frac{50x}{16x^4y^9}&=\frac{25x}{8x^4y^9}\\ \end{aligned}$$

I have completed the problem several times consistently receiving the same answer which I believe to be correct; although, I plugged the expression into mathway.com/algebra and I got a different answer. I wanted to see if someone could give the correct answer and explain their steps for how they were able to arrive at it.

Answer 1

You can do it all in one fell swoop. To start with, forget about the constant factors and work with the variables only (remember, add exponents from numerator factors and subtract exponents from denominator factors; also, $x=x^1$ and $y=y^1$):

$$x^{4-3+2-5+4-5}y^{1-3+1-7+2-3}=x^{-3}y^{-9} = \frac1{x^3y^9}$$

For the constant factors, I notice everything has factors of $2$ or $5$, so I will factor them and do the same thing. Note $10=2\cdot 5$, $25=5^2$, $8=2^3$,

$$2^{1-3-1}5^{1-1+2}=2^{-3}5^{2}=\frac{25}{8}$$ Putting it together, you have $$\boxed{\frac{25}{8x^3y^9}}$$ Of course, you could have handled the constants and the variables at the same time. I just broke it into two steps for convenience.

Note: In your answer, remember that the $x$ in the numerator is really $x^1$, so you can combine it with the $x^4$ in the denominator to simplify a little more.